A categorification of quantum sl(2)

TL;DR

This paper constructs a 2-category that categorifies Lusztig's quantum sl(2), providing a graphical calculus framework that lifts algebraic structures to a categorical level, and offers representations for each positive integer N.

Contribution

It introduces a novel graphical calculus-based 2-category that categorifies Lusztig's quantum sl(2) and its irreducible representations using iterated flag varieties.

Findings

The split Grothendieck ring of the 2-category is isomorphic to Lusztig's algebra.

Indecomposable morphisms lift Lusztig's canonical basis.

Constructed representations categorify irreducible quantum sl(2) modules.

Abstract

We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for each positive integer N. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).